1. 7.4 Properties of Exponents

2. Perform the indicated operations. 1. (2x3y6)(3x8y7) 2. (–2xy)3
Example: Performing Operations
Perform the indicated operations.
1. (2x3y6)(3x8y7) (–2xy)3

3. Solution 1. 2.

4. Simplifying a Power Expression
A power expression is simplified if
1. It includes no parentheses.
2. In any monomial, each variable or constant appears as a base at most once. For example, for nonzero x, we write x3x5 = x8.
3. Each numerical expression (such as 52) has been calculated, and each numerical fraction has been simplified.

5. Quotient Property for Exponents
If m and n are counting numbers and x is nonzero, then
In words: To divide two powers of x, keep the base and subtract the exponents.

6. Example: Quotient Property
Simplify.

7. Solution 1.

8. Solution 2.

9. Zero as an Exponent
What is the meaning of 20? Computing powers of 2 can suggest the meaning:
Each time we decrease the exponent by 1, the value is divided by 2. This pattern suggests that 20 = 1.

10. Zero Exponent
Definition
For nonzero x, x0 = 1

11. Raising a Quotient to a Power
If n is counting number and y is nonzero, then
In words: To raise a quotient to a power, raise both the numerator and the denominator to the power.

12. Example: Raising a Quotient to a Power
Simplify.

13. Solution

14. Raising a Power to a Power
If m and n are counting numbers, then
In words: To raise a power to a power, keep the base and multiply the exponents.

15. Example: Raising a Power to a Power
Simplify.

16. Solution

17. Example: Simplifying Power Expressions1. 2. 3.

18. Solution 1. 2.

19. Solution 3.

20. Example: Simplifying Power Expressions

21. Solution 1.

22. Solution 2.